Chapter 8 The Schwarzschild Solution 8.1 Cartan’s structure equations 8.1.1 Curvature forms This section deals with a generalisation of the connection coefficients, and the torsion and curvature tensor components, to arbitrary bases. This will prove enormously efficient in our further discussion of the Schwarzschild solution. Let M be a differentiable manifold, {e i } an arbitrary basis for vector fields and {θ i } an arbitrary basis for dual vector fields, or 1-forms. Connection forms In analogy to the Christoffel symbols, we introduce the connection forms by ∇ v e i = ω j i (v)e j . (8.1) Since ∇ v e i is a vector, ω j i (v) ∈ R is a real number, and thus ω j i ∈ 1 is a dual vector, or a one-form. Since, by definition (3.2) of the Christoffel symbols ∇ ∂ k ∂ j =Γ i kj ∂ i = ω i j (∂ k )∂ i (8.2) in the coordinate basis {∂ i }, we have in that particular basis, ω i j =Γ i kj dx k . (8.3) Since θ i , e j is a constant (which is either zero or unity if the basis is orthonormal), we must have 0 = ∇ v θ i , e j = ∇ v θ i , e j + θ i , ∇ v e j = ∇ v θ i , e j + θ i ,ω k j (v)e k = ∇ v θ i , e j + ω i j (v) . (8.4) 111
Transcript
Chapter 8
The Schwarzschild Solution
8.1 Cartan’s structure equations
8.1.1 Curvature forms
This section deals with a generalisation of the connection coefficients,and the torsion and curvature tensor components, to arbitrary bases.This will prove enormously efficient in our further discussion of theSchwarzschild solution.
Let M be a differentiable manifold, {ei} an arbitrary basis for vector fieldsand {θi} an arbitrary basis for dual vector fields, or 1-forms.
Connection forms
In analogy to the Christoffel symbols, we introduce the connectionforms by
∇vei = ωji (v)e j . (8.1)
Since ∇vei is a vector, ω ji (v) ∈ R is a real number, and thus ω j
i ∈∧1 is
a dual vector, or a one-form.
Since, by definition (3.2) of the Christoffel symbols
∇∂k∂ j = Γik j ∂i = ω
ij(∂k)∂i (8.2)
in the coordinate basis {∂i}, we have in that particular basis,
ωij = Γ
ik j dxk . (8.3)
Since 〈θi, e j〉 is a constant (which is either zero or unity if the basis isorthonormal), we must have
0 = ∇v〈θi, e j〉 = 〈∇vθi, e j〉 + 〈θi,∇ve j〉= 〈∇vθi, e j〉 + 〈θi, ωk
j(v)ek〉= 〈∇vθi, e j〉 + ωi
j(v) . (8.4)
111
112 8 The Schwarzschild Solution
From this result, we can conclude
∇vθi = −ωij(v)θ
j (8.5)
for the covariant derivative of θi in the direction of v. Without specifyingthe vector v, we find the covariant derivative
∇θi = −θ j ⊗ ωij . (8.6)
Let now α ∈∧1 be a one-form such that α = αiθ
i with arbitrary functionsαi. Then, the equations we have derived so far imply
where we have used the differential of the function αi, defined in (2.35)by dαi(v) = v(αi), together with the notation 〈w, v〉 = w(v) for a vector vand a dual vector w. More generally, this expression can be written asthe covariant derivative
∇α = θi ⊗ (dαi − αkωki ) . (8.8)
Similarly, for a vector field x = xiei, we find
∇vx = 〈dxi + xkωik, v〉ei (8.9)
or∇x = ei ⊗ (dxi + ωi
k xk) (8.10)
for the covariant derivative of the vector x.
?
Derive the expressions (8.9) and(8.10) yourself, beginning with(8.1).
8.1.2 Torsion and curvature forms
We are now in a position to use the connection forms for defining thetorsion and curvature forms.
Torsion and curvature forms
By definition, the torsion T (x, y) is a vector, which can be written interms of the torsion forms Θi as
T (x, y) = Θi(x, y)ei . (8.11)
Obviously, Θi ∈∧2 is a two-form, such that Θi(x, y) ∈ R is a real
number.In the same manner, we express the curvature by the curvature formsΩi
j ∈∧2,
R(x, y)e j = Ωij(x, y)ei . (8.12)
The next important step is now to realise that the torsion and curvature2-forms satisfy Cartan’s structure equations:
8.1 Cartan’s structure equations 113
Figure 8.1 Élie Cartan (1869–1951), French mathematician. Source:Wikipedia
Cartan’s structure equations
In terms of the connection forms ωij, the torsion forms Θi and the
curvature forms Ωij are determined by Cartan’s structure equations,
Θi = dθi + ωij ∧ θ j
Ωij = dωi
j + ωik ∧ ω
kj . (8.13)
Their proof is straightforward. To prove the first structure equation, weinsert the definition (3.45) of the torsion to obtain as a first step
According to (5.66), the first term can be expressed by the exteriorderivative of the θi, and since the second term is antisymmetric in x andy, we can write this as
from which the first structure equation follows immediately.
The proof of the second structure equation proceeds similarly, using thedefinition (3.51) of the curvature. Thus,
Ωij(x, y)ei = ∇x∇ye j − ∇y∇xe j − ∇[x,y]e j
= ∇x(ωij(y)ei) − ∇y(ωi
j(x)ei) − ωij([x, y])ei
= xωij(y)ei + ω
ij(y)∇xei
− yωij(x)ei − ωi
j(x)∇yei − ωij([x, y])ei
=[xωi
j(y) − yωij(x) − ωi
j([x, y])]
ei
+[ωi
j(y)ωki (x) − ωi
j(x)ωki (y)
]ek
= dωij(x, y)ei + (ωk
i ∧ ωij)(x, y)ek , (8.17)
which proves the second structure equation.
?
Carry out all steps of the deriva-tions (8.15) and (8.17) yourselfand convince yourself that theyare correct. Now, let us use the curvature forms Ωi
j to define tensor components Rijkl
by
Ωij ≡
12
Rijkl θ
k ∧ θl , (8.18)
whose antisymmetry in the last two indices is obvious by definition,
Rijkl = −Ri
jlk . (8.19)
In an arbitrary basis {ei}, we then have
〈θi, R(ek, el)e j〉 = 〈θi,Ωsj(ek, el)es〉 = Ωi
j(ek, el) = Rijkl . (8.20)
Comparing this to the components of the curvature tensor in the coordi-nate basis {∂i} given by (3.56) shows that the functions Ri
jkl are indeedthe components of the curvature tensor in the arbitrary basis {ei}.
A similar operation shows that the functions T ijk defined by
Θi ≡ 12
T ijk θ
j ∧ θk (8.21)
are the elements of the torsion tensor in the basis {ei}, since
〈θi,T (e j, ek)〉 = 〈θi,Θs(e j, ek)es〉 = Θi(e j, ek) = T ijk . (8.22)
Thus, Cartan’s structure equations allow us to considerably simplify thecomputation of curvature and torsion for an arbitrary metric, providedwe find a base in which the metric appears simple (e.g. diagonal andconstant).
8.2 Stationary and static spacetimes 115
Symmetry of the connection forms
We mention without proof that the connection ∇ is metric if and only if
ωi j + ω ji = dgi j , (8.23)
where the definitions
ωi j ≡ gikωkj and gi j ≡ g(ei, e j) (8.24)
were used, i.e. the gi j are the components of the metric in the arbitrarybasis {ei}.
8.2 Stationary and static spacetimes
Stationary spacetimes (M, g) are defined to be spacetimes which havea time-like Killing vector field K. This means that observers movingalong the integral curves of K do not notice any change.
?
What exactly are Killing vectorfields? How are they defined, andwhat do they mean?This definition implies that we can introduce coordinates in which the
components gμν of the metric do not depend on time. To see this, supposewe choose a space-like hypersurface Σ ⊂ M and construct the integralcurves of K through Σ.
We further introduce arbitrary coordinates on Σ and carry them into Mas follows: let φt be the flow of K, p0 ∈ Σ and p = φt(p0), then thecoordinates of p are chosen as (t, x1(p0), x2(p0), x3(p0)). These are theso-called Lagrange coordinates of p.
In these coordinates, K = ∂0, i.e. Kμ = δμ0. From the derivation of theKilling equation (5.34), we further have that the components of the Liederivative of the metric are
(LKg)μν = Kλ∂λgμν + gλν∂μKλ + gμλ∂νKλ
= ∂0gμν = 0 , (8.25)
which proves that the gμν do not depend on time in these so-calledadapted coordinates.
We can straightforwardly introduce a one-form ω corresponding to theKilling vector K by ω = K�. This one-form obviously satisfies
ω(K) = 〈K,K〉 � 0 . (8.26)
Suppose that we now have a stationary spacetime in which we haveintroduced adapted coordinates and in which also g0i = 0. Then, theKilling vector field is orthogonal to the spatial sections, for which t =const. Then, the one-form ω is quite obviously
ω = g00 cdt = 〈K,K〉 cdt , (8.27)
116 8 The Schwarzschild Solution
because K = ∂0. This then trivially implies the Frobenius condition
ω ∧ dω = 0 (8.28)
because the exterior derivative d satisfies d ◦ d ≡ 0.
Conversely, it can be shown that if the Frobenius condition holds, the one-form ω can be written in the form (8.27). For a vector field v tangentialto a spacelike section defined by t = const., we have
because t = const., and thus K is then perpendicular to the spatial section.Thus, K = ∂0 and
g0i = 〈∂0, ∂i〉 = 〈K, ∂i〉 = 0 . (8.30)
Stationary and static spacetimes
Thus, in a stationary spacetime with time-like Killing vector field K,the Frobenius condition (8.28) for the one-form ω = K� is equivalentto the condition g0i = 0 in adapted coordinates. Such spacetimes arecalled static. In other words, stationary spacetimes are static if and onlyif the Frobenius condition holds.
In static spacetimes, the metric can thus be written in the form
g = g00(�x )c2dt2 + gi j(�x )dxidx j . (8.31)
8.3 The Schwarzschild solution
8.3.1 Form of the metric
Formally speaking, the Schwarzschild solution is a static, sphericallysymmetric solution of Einstein’s field equations for vacuum spacetime.
From our earlier considerations, we know that a static spacetime is astationary spacetime whose (time-like) Killing vector field satisfies theFrobenius condition (8.28).
As the spacetime is (globally) stationary, we know that we can introducespatial hypersurfaces Σ perpendicular to the Killing vector field which, inadapted coordinates, is K = ∂0. The manifold (M, g) can thus be foliatedas M = R × Σ.
?
How does a product space com-posed of two manifolds attainsthe structure of a product mani-fold. From (8.31), we then know that, also in adapted coordinates, the metric
acquires the formg = −φ2 c2dt2 + h , (8.32)
8.3 The Schwarzschild solution 117
where φ is a smoothly varying function on Σ and h is the metric ofthe spatial sections Σ. Under the assumption that K is the only time-like Killing vector field which the spacetime admits, t is a uniquelydistinguished time coordinate, and we can write
− φ2 = 〈K,K〉 . (8.33)
The stationarity of the spacetime, expressed by the existence of the singleKilling vector field K, thus allows a convenient foliation of the spacetimeinto spatial hypersurfaces or foils Σ and a time coordinate.
Furthermore, the spatial hypersurfaces Σ are expected to be sphericallysymmetric. This means that the group SO(3) (i.e. the group of rotationsin three dimensions) must be an isometry group of the metric h. Theorbits of SO(3) are two-dimensional, space-like surfaces in Σ. Thus,SO(3) foliates the spacetime (Σ, h) into invariant two-spheres.
Let the surface of these two-spheres be A, then we define a radial coordi-nate for the Schwarzschild metric requiring
4πr2 = A (8.34)
as in Euclidean geometry. Moreover, the spherical symmetry impliesthat we can introduce spherical polar coordinates (ϑ, ϕ) on one partic-ular orbit of SO(3) which can then be transported along geodesic linesperpendicular to the orbits. Then, the spatial metric h can be written inthe form
h = e2b(r)dr2 + r2(dϑ2 + sin2 ϑdϕ2
), (8.35)
where the exponential factor was introduced to allow a scaling of theradial coordinate.
?
Why could it be useful to repre-sent the coefficient of dr2 in h byan exponential?Due to the stationarity of the metric and the spherical symmetry of the
spatial sections, 〈K,K〉 can only depend on r. We set
φ2 = −〈K,K〉 = e2a(r) . (8.36)
The full metric g is thus characterised by two radial functions a(r) andb(r) which we need to determine. The exponential functions in (8.35)and (8.36) are chosen to ensure that the prefactors ea and eb are alwayspositive.
The spatial sections Σ are now foliated according to
Σ = I × S 2 , I ⊂ R+ , (8.37)
with coordinates r ∈ I and (ϑ, ϕ) ∈ S 2.
118 8 The Schwarzschild Solution
Metric for static, spherically-symmetric spacetimes
In the Schwarzschild coordinates (t, r, ϑ, ϕ), the metric of a static,spherically-symmetric spacetime has the form
The functions a(r) and b(r) are constrained by the requirement that themetric should asymptotically turn flat, which means
a(r) → 0 , b(r) → 0 for r → ∞ . (8.39)
They must be determined by inserting the metric (8.38) into the vacuumfield equations, G = 0.
8.3.2 Connection and curvature forms
In order to evaluate Einstein’s field equations for the Schwarzschildmetric, we now need to compute the Riemann, Ricci, and Einsteintensors. Traditionally, one would begin this step with computing allChristoffel symbols of the metric (8.38). This very lengthy and error-prone procedure can be considerably shortened using Cartan’s structureequations (8.13) for the torsion and curvature forms Θi and Ωi
j.
To do so, we need to introduce a suitable basis, or tetrad {ei}, or alter-natively a dual tetrad {θi}. Guided by the form of the metric (8.38), wechoose
Obviously, dg = 0, and thus (8.23) implies that the connection forms ωμνneed to be antisymmetric,
ωμν = −ωνμ . (8.42)
Given the dual tetrad {θμ}, we must take their exterior derivatives. Forthis purpose, we apply the expression (??) and find, for dθ0,
dθ0 = dea ∧ cdt = −a′ea cdt ∧ dr . (8.43)
because dea = a′ea dr. Similarly, we find
dθ1 = 0 (8.44)
8.3 The Schwarzschild solution 119
because dr ∧ dr = 0, further
dθ2 = dr ∧ dϑ (8.45)
anddθ3 = sinϑ dr ∧ dϕ + r cosϑ dϑ ∧ dϕ . (8.46)
Using (8.40), we can also express the coordinate differentials by the dualtetrad,
cdt = e−aθ0 , dr = e−bθ1 , dϑ =θ2
r, dϕ =
θ3
r sinϑ, (8.47)
so that we can write the exterior derivatives of the dual tetrad as
dθ0 = a′e−b θ1 ∧ θ0 , dθ1 = 0 , dθ2 =e−b
rθ1 ∧ θ2 ,
dθ3 =e−b
rθ1 ∧ θ3 + cotϑ
rθ2 ∧ θ3 . (8.48) ?
Test by independent calculationwhether you can confirm the dif-ferentials (8.48).Since the torsion must vanish, Θi = 0, Cartan’s first structure equation
from (8.13) impliesdθ μ = −ωμν ∧ θν . (8.49)
Connection forms
With (8.48), this suggests that the connection forms of a static,spherically-symmetric metric are
ω01 = ω
10 =
a′θ0
eb , ω02 = ω
20 = 0 , ω0
3 = ω30 = 0 ,
ω21 = −ω1
2 =θ2
reb , ω31 = −ω
13 =θ3
reb ,
ω32 = −ω
23 =
cotϑ θ3
r. (8.50)
They satisfy the antisymmetry condition (8.42) and Cartan’s first struc-ture equation (8.49) for a torsion-free connection.
?
Why can none of the connectionforms in (8.50) depend on ϕ?
For evaluating the curvature forms Ωμν , we first recall that the exteriorderivative of a one-form ω multiplied by a function f is
where we have used (8.47) and (8.48) and abbreviated A := (a′′ − a′b′ +a′2)E with E := exp(−2b).
In much the same way and using this definition of E, we find
dω21 = −
b′Erθ1 ∧ θ2 ,
dω31 = −
b′Erθ1 ∧ θ3 + cotϑ
r2eb θ2 ∧ θ3 ,
dω32 = −
1r2 θ
2 ∧ θ3 . (8.53)
This yields the curvature two-forms according to (8.13).
Curvature forms of a static, spherically-symmetric metric
The curvature forms of a static, spherically-symmetric metric are
Ω01 = dω0
1 = −A θ0 ∧ θ1 = Ω10
Ω02 = ω
01 ∧ ω
12 = −
a′Erθ0 ∧ θ2 = Ω2
0
Ω03 = ω
01 ∧ ω
13 = −
a′Erθ0 ∧ θ3 = Ω3
0
Ω12 = dω1
2 + ω13 ∧ ω3
2 =b′E
rθ1 ∧ θ2 = −Ω2
1
Ω13 = dω1
3 + ω12 ∧ ω2
3 =b′E
rθ1 ∧ θ3 = −Ω3
1
Ω23 = dω2
3 + ω21 ∧ ω1
3 =1 − E
r2 θ2 ∧ θ3 = −Ω32 . (8.54)
?
Perform the calculations lead-ing to the curvature forms (8.54)yourself and see whether you canconfirm them.
The remaining curvature two-forms follow from antisymmetry since
Ωμν = gμλΩλν = −Ωνμ , (8.55)
because of the (anti-)symmetries of the curvature.
8.4 Solution of the field equations
8.4.1 Components of the Ricci and Einstein tensors
The components of the curvature tensor are given by (8.20), and thus thecomponents of the Ricci tensor in the tetrad {eα} are
Rμν = Rλμλν = Ωλμ(eλ, eν) . (8.56)
Thus, the components of the Ricci tensor in the Schwarzschild tetrad are
R00 = Ω10(e1, e0) + Ω2
0(e2, e0) + Ω30(e3, e0) = A +
2a′Er,
R11 = Ω01(e0, e1) + Ω2
1(e2, e1) + Ω31(e3, e1) = −A +
2b′Er
(8.57)
8.4 Solution of the field equations 121
and, with B := (b′ − a′)E/r,
R22 = Ω02(e0, e2) + Ω1
2(e1, e2) + Ω32(e3, e2) =: B +
1 − Er2
R33 = Ω03(e0, e3) + Ω1
3(e1, e3) + Ω23(e2, e3) = R22 (8.58)
The Ricci scalar becomes
R = −2A + 4B + 21 − E
r2 , (8.59)
such that we can now determine the components of the Einstein tensorin the tetrad {eα} :
Einstein tensor for a static, spherically-symmetric metric
The Einstein tensor of a static, spherically-symmetric metric has thecomponents
G00 = R00 −R2g00 =
1r2 − E
(1r2 −
2b′
r
)
G11 = −1r2 + E
(1r2 +
2a′
r
)G22 = E (A − B) = G33 . (8.60)
All off-diagonal components of Gμν vanish identically.
?
Convince yourself of the compo-nents (8.60) of the Einstein ten-sor.
8.4.2 The Schwarzschild metric
The vacuum field equations now require that all components of theEinstein tensor vanish. In particular, then,
0 = G00 +G11 =2Er
(a′ + b′) (8.61)
shows that a′ + b′ = 0. Since a + b → 0 asymptotically for r → ∞,integrating a + b from r → ∞ indicates that a + b = 0 everywhere, orb = −a.
After multiplying with r2, equation G00 = 0 itself implies that
E(1 − 2rb′) = 1 ⇔ (rE)′ = 1 . (8.62)
Therefore, (8.62) is equivalent to
rE = r +C ⇔ E = 1 +Cr, (8.63)
with an integration constant C to be determined.
122 8 The Schwarzschild Solution
Figure 8.2 Karl Schwarzschild (1873–1916), German astronomer andphysicist. Source: Wikipedia
Since a = −b, this also allows to conclude that
e2a = E = 1 +Cr. (8.64)
The integration constant C is finally determined by the Newtonian limit.We have seen before in (4.80) that the 0-0 element of the metric must berelated to the Newtonian gravitational potential as g00 = −(1 + 2Φ/c2) inorder to meet the Newtonian limit. The Newtonian potential of a pointmass M at a distance r is
Φ = −GMr. (8.65)
Together with (8.64), this shows that the Newtonian limit is reached bythe Schwarzschild solution if the integration constant C is set to
C = −2GMc2 =: −2m with m =
GMc2 ≈ 1.5 km
(MM
). (8.66)
Schwarzschild metric
We thus obtain the Schwarzschild solution for the metric,
ds2 = −(1 − 2m
r
)c2dt2 +
dr2
1 − 2mr
+ r2(dϑ2 + sin2 ϑdϕ2
). (8.67)
The Schwarzschild metric (8.67) has an (apparent) singularity at r = 2mor
r = Rs ≡2GM
c2 , (8.68)
8.4 Solution of the field equations 123
the so-called Schwarzschild radius. We shall clarify the meaning of thissingularity later.
In order to illustrate the geometrical meaning of the spatial part ofthe Schwarzschild metric, we need to find a geometrical interpretationfor its radial dependence. Specialising to the equatorial plane of theSchwarzschild solution, ϑ = π/2 and t = 0, we find the induced spatialline element
dl2 =dr2
1 − 2m/r+ r2dϕ2 (8.69)
on that plane.
On the other hand, consider a surface of rotation in the three-dimensionalEuclidean space E3. If we introduce the adequate cylindrical coordinates(r, φ, z) on E3 and rotate a curve z(r) about the z axis, we find the inducedline element
dl2 = dz2 + dr2 + r2dϕ2 =
(dzdr
)2
dr2 + dr2 + r2dϕ2
= (1 + z′2)dr2 + r2dϕ2 . (8.70)
We can now try and identify the two induced line elements from (8.69)and (8.70) and find that this is possible if
z′ =(
11 − 2m/r
− 1)1/2
=
√2m
r − 2m, (8.71)
which is readily integrated to yield
z =√
8m(r − 2m) + const. or z2 = 8m(r − 2m) , (8.72)
if we set the integration constant to zero.
This shows that the geometry on the equatorial plane of the spatialsection of the Schwarzschild solution can be identified with a rotationalparaboloid in E3. In other words, the dependence of radial distances onthe radius r is equivalent to that on a rotational paraboloid (cf. Fig. 8.3).
8.4.3 Birkhoff’s theorem
Suppose now we had started from a spherically symmetric vacuumspacetime, but with explicit time dependence of the functions a andb, such that the spacetime could either expand or contract. Then, arepetition of the derivation of the connection and curvature forms, and
124 8 The Schwarzschild Solution
Figure 8.3 Surface of rotation illustrating the spatial part of the Schwarz-schild metric.
the components of the Einstein tensor following from them, had resultedin the new components Gμν
G00 = G00 , G11 = G11
G22 = G22 − e−2a(b2 − ab − b
)= G33
G10 =2br
e−a−b (8.73)
and Gμν = 0 for all other components.
The vacuum field equations imply G10 = 0 and thus b = 0, hence b mustbe independent of time. From G00 = 0, we can again conclude (8.63),i.e. b retains the same form as before. Similarly, since G00 + G11 =
G00 +G11, the requirement a′ + b′ = 0 must continue to hold, but nowthe time dependence of a allows us to conclude only that
a = −b + f (t) , (8.74)
where f (t) is an otherwise unconstrained function of time only. Thus,the line element then reads
ds2 = −e2 f
(1 − 2m
r
)c2dt2 + dl2 , (8.75)
where dl2 is the unchanged line element of the spatial sections.
Introducing the new time coordinate t′ by
t′ =∫
e f dt (8.76)
converts (8.75) back to the original form (8.67) of the Schwarzschildmetric.
8.4 Solution of the field equations 125
Birkhoff’s theorem
This is Birkhoff’s theorem, which states that a spherically symmetricsolution of Einstein’s vacuum equations is necessarily static for r > 2m.
